Optimum control of deterministic and stochastic distributed populations with intra-species competition†

Abstract

Abstract Most of the literature devoted to the study of dynamic populations with predators is focused on the use of the so-called ‘population law’ discovered by Verhulst in 1937, and generalized later by several authors. Basically this model deals with the total number of individuals in the population, irrespective of their distribution over the geographical space of definition. In contrast, the present paper suggests a distributed approach to populations with intra-species competitions, which explicitly takes into account the distributed parameter, for instance the age, of the individuals. Two models are given: a distributed version of the logistic law and a generalization of the transfer equation, and two frameworks are considered: the deterministic case and the stochastic one. The equations so obtained are mainly distributed differential integral equations; one considers The optimum control or such systems and necessary optimality conditions are given. The covariance of the deviation from a steady state in the presence of random fluctuations is derived. This model applies not only to biological populations, but also to social and economic systems, and it may be used as a basis for general studies on populations with intra-species competitions.